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added restriction $a\ne0$
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Arnold Neumaier
Arnold Neumaier
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I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for the space of rational functions of a single complex variable of arbitrary, unbounded denominator degree (and, if needed, a finite, or zero, limit for $z\to\infty$). Or spaces of meromorphic functions containing these functions.

The main requirement is that any sequence of functions $r_n$ defined by $r_n(z):=a_n/(b_n-z)$ for $z$-independent $a_n\to a$$a_n\to a\ne 0$ and $b_n\to b$ converges to to the function $r$ defined by $r(z):=a/(b-z)$.

If possible, I'd like to have a metric which gives a nice formula for the distance of $a/(b-z)$ to $a'/(b'-z)$.

I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for the space of rational functions of a single complex variable of arbitrary, unbounded denominator degree (and, if needed, a finite limit for $z\to\infty$). Or spaces of meromorphic functions containing these functions.

The main requirement is that any sequence of functions $r_n$ defined by $r_n(z):=a_n/(b_n-z)$ for $z$-independent $a_n\to a$ and $b_n\to b$ converges to to the function $r$ defined by $r(z):=a/(b-z)$.

If possible, I'd like to have a metric which gives a nice formula for the distance of $a/(b-z)$ to $a'/(b'-z)$.

I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for the space of rational functions of a single complex variable of arbitrary, unbounded denominator degree (and, if needed, a finite, or zero, limit for $z\to\infty$). Or spaces of meromorphic functions containing these functions.

The main requirement is that any sequence of functions $r_n$ defined by $r_n(z):=a_n/(b_n-z)$ for $z$-independent $a_n\to a\ne 0$ and $b_n\to b$ converges to to the function $r$ defined by $r(z):=a/(b-z)$.

If possible, I'd like to have a metric which gives a nice formula for the distance of $a/(b-z)$ to $a'/(b'-z)$.

added details to the specification
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Arnold Neumaier
Arnold Neumaier
  • 3.9k
  • 17
  • 37

I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for spacesthe space of rational (or meromorphic) functions of a single complex variable of arbitrary, unbounded denominator degree (and, if needed, a finite limit for $z\to\infty$). Or spaces of meromorphic functions containing these functions.

The main requirement is that any sequence of functions $r_n$ defined by $r_n(z):=a_n/(b_n-z)$ for $z$-independent $a_n\to a$ and $b_n\to b$ converges to to the function $r$ defined by $r(z):=a/(b-z)$.

If possible, I'd like to have a metric which gives a nice formula for the distance of $a/(b-z)$ to $a'/(b'-z)$.

I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for spaces of rational (or meromorphic) functions of a single complex variable.

The main requirement is that any sequence of functions $r_n$ defined by $r_n(z):=a_n/(b_n-z)$ for $z$-independent $a_n\to a$ and $b_n\to b$ converges to to the function $r$ defined by $r(z):=a/(b-z)$.

I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for the space of rational functions of a single complex variable of arbitrary, unbounded denominator degree (and, if needed, a finite limit for $z\to\infty$). Or spaces of meromorphic functions containing these functions.

The main requirement is that any sequence of functions $r_n$ defined by $r_n(z):=a_n/(b_n-z)$ for $z$-independent $a_n\to a$ and $b_n\to b$ converges to to the function $r$ defined by $r(z):=a/(b-z)$.

If possible, I'd like to have a metric which gives a nice formula for the distance of $a/(b-z)$ to $a'/(b'-z)$.

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Arnold Neumaier
Arnold Neumaier
  • 3.9k
  • 17
  • 37

Natural topologies for the space of rational functions

I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for spaces of rational (or meromorphic) functions of a single complex variable.

The main requirement is that any sequence of functions $r_n$ defined by $r_n(z):=a_n/(b_n-z)$ for $z$-independent $a_n\to a$ and $b_n\to b$ converges to to the function $r$ defined by $r(z):=a/(b-z)$.